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arXiv · 2005.02110

Higher Specht bases for generalizations of the coinvariant ring

Maria Gillespie·Brendon Rhoades
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Abstract

The classical coinvariant ring $R_n$ is defined as the quotient of a polynomial ring in $n$ variables by the positive-degree $S_n$-invariants. It has a known basis that respects the decomposition of $R_n$ into irreducible $S_n$-modules, consisting of the higher specht polynomials due to Ariki, Terasoma, and Yamada. We provide an extension of the higher Specht basis to the generalized coinvariant rings $R_{n,k}$. We also give a conjectured higher Specht basis for the Garsia-Procesi modules $R_\mu$, and provide a proof of the conjecture in the case of two-row partition shapes $\mu$. We then combine these results to give a higher Specht basis for an infinite subfamily of the modules $R_{n,k,\mu}$ recently defined by Griffin, which are a common generalization of $R_{n,k}$ and $R_{\mu}$.

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Maria Gillespie, Brendon Rhoades. 2020-05-05. Higher Specht bases for generalizations of the coinvariant ring. https://arxiv.org/abs/2005.02110

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